Abstract
For a>0, let Wa1(t) and Wa2(t) be the a-neighbourhoods of two independent standard Brownian motions in Rd starting at 0 and observed until time t. We prove that, for d≥3 and c>0,
limt→∞1t(d−2)/dlogP(|Wa1(ct)∩Wa2(ct)|≥t)=−Iκad(c)
and derive a variational representation for the rate constant Iκad(c). Here, κa is the Newtonian capacity of the ball with radius a. We show that the optimal strategy to realise the above large deviation is for Wa1(ct) and Wa2(ct) to “form a Swiss cheese”: the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t1/d according to a certain optimal profile.
We study in detail the function c↦Iκad(c). It turns out that Iκad(c)=Θd(κac)/κa, where Θd has the following properties: (1) For d≥3: Θd(u)<∞ if and only if u∈(u⋄,∞), with u⋄ a universal constant; (2) For d=3: Θd is strictly decreasing on (u⋄,∞) with a zero limit; (3) For d=4: Θd is strictly decreasing on (u⋄,∞) with a nonzero limit; (4) For d≥5: Θd is strictly decreasing on (u⋄,ud) and a nonzero constant on [ud,∞), with ud a constant depending on d that comes from a variational problem exhibiting "leakage". This leakage is interpreted as saying that the two Wiener sausages form their intersection until time c∗t, with c∗=ud/κa, and then wander off to infinity in different directions. Thus, c∗ plays the role of a critical time horizon in d≥5.
We also derive the analogous result for d=2, namely,
limt→∞1logtlogP(|Wa1(ct)∩Wa2(ct)|≥t/logt)=−I2π2(c),
where the rate constant has the same variational representation as in d≥3 after κa is replaced by 2π. In this case I2π2(c)=Θ2(2πc)/2π with Θ2(u)<∞ if and only if u∈(u⋄,∞) and Θ2 is strictly decreasing on (u⋄,∞) with a zero limit.

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